There are several ways we could go about obtaining new series from these ones. First, we can substitute inside an existing series. Then, we could differentiate or integrate existing series. We'll consider these with some examples.

Substituting Inside a Taylor Series

If we're given one series for a function and we wish to create the series for , then we can replace for in the Taylor series of .

Examples:

Find the Taylor series for about .

Solution: We plug in everywhere we see in the Taylor series expansion of about . That is, the desired Taylor series is

Find the Taylor series for about .

Solution: We know that

If we replace the in this series by , we will then have the desired series. So, we have

Example 2 on page 520 provides a much more complicated example.

Differentiation and Integration

If you know a Taylor series for a function, you can find the Taylor series for its derivative or integral by differentiating or integrating the terms of the Taylor series. A few examples make this more obvious.

Examples:

Find the Taylor series for around zero.

Solution: We know the Taylor series for and we know that

Now, we have

Find the Taylor series for about . Use the resulting Taylor series to approximate .

Solution: We know that

Thus, if we can find the Taylor series for this function, we can integrate it and get back to . We can perform a substitution and find

Then, we integrate and find

So, we now have a Taylor series for around . Approxmating is then a simple trick. We recall that